To go even deeper, using your example...

What counts as "injured" or "killed"? If shots are fired and the resulting human stampede kills 4 people, does that count as 4 mass shooting deaths? Obviously the shooter is at fault, but these details affect the interpretation of events.

My undergrad was in statistics. In our capstone course, my professor had us read journal articles and discuss the statistical analyses within. I remember one study we read (peer reviewed, a couple dozen citations), and my professor's take away was, "I can't say it's wrong, but based on the data they gave, I can't for the life of me figure out how they reached their statistical conclusions." So yeah, it's a "fact" that the researchers reached a certain conclusion, but the conclusion itself is not fact.

I don't believe in post-truth, but "Facts are facts; truth is truth" is a philosophical statement, not a practical one. And even then, we have an entire field of philosophy to iron out those details, which we call epistemology.

In this example, the empirical facts are "hundreds of incidents where four or more people are injured" and "no incidents where twenty or more people are killed". Those facts still exist. The different definitions of "mass shooting" are spin, which obscures facts, but does not eliminate them. Yes, it is hard to pierce the spin to find the facts, but the facts are there somewhere.

I just read the short book "How To Lie With Statistics" this year and it holds up incredibly well despite being nearly 70 years old!

You're making a strong statement in favor of fact checking.

People can lie with statistics and people can lie without statistics. The latter is much easier, but the former is possible, as you lay out.

That's why we need to check whether an alleged fact is true, or at least can be confirmed from multiple sources of evidence so it can be accepted as true for the time being. We can also check statistics for anomalies and errors. Statisticians do that all the time.

All of that is fact checking.